Pontryagin duality

1 Pontryagin dual

Let G be a locally compact abelian groupMathworldPlanetmath (http://planetmath.org/TopologicalGroup) and 𝕋 the 1-torus (http://planetmath.org/NTorus), i.e. the unit circle in β„‚.

Definition - A continuousPlanetmathPlanetmath homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath GβŸΆπ•‹ is called a characterPlanetmathPlanetmath of G. The set of all characters is called the Pontryagin dual of G and is denoted by G^.

Under pointwise multiplication G^ is also an abelian group. Since G^ is a group of functions we can make it a topological groupMathworldPlanetmath under the compact-open topologyMathworldPlanetmath (topologyMathworldPlanetmath of convergence on compact sets).

2 Examples

  • β€’

    β„€^≅𝕋, via n↦zn with zβˆˆπ•‹.

  • β€’

    𝕋^β‰…β„€, via z↦zn with nβˆˆβ„€.

  • β€’

    ℝ^≅ℝ, via t↦ei⁒s⁒t with sβˆˆβ„.

3 Properties

The following are some important of the dual group:

Theorem - Let G be a locally compact abelian group. We have that

  • β€’

    G^ is also locally compact.

  • β€’

    G^ is second countable if and only if G is second countable.

  • β€’

    G^ is compact if and only if G is discrete.

  • β€’

    G^ is discrete if and only if G is compact.

  • β€’

    (βŠ•i∈JGi)^β‰…βŠ•i∈JGi^ for any finite setMathworldPlanetmath J. This isomorphismMathworldPlanetmathPlanetmath is natural.

4 Pontryagin duality

Let f:G⟢H be a continuous homomorphism of locally compact abelian groups. We can associate to it a canonical map f^:H^⟢G^ defined by


This canonical construction preserves identity mappings and compositions, i.e. the dualization process ^ is a functorMathworldPlanetmath:

Theorem - The dualization ^:π‹πœπ€βŸΆπ‹πœπ€ is a contravariant functor from the categoryMathworldPlanetmath of locally compact abelian groups to itself.

5 Isomorphism with the second dual

Although in general there is not a canonical identification of G with its dual G^, there is a natural isomorphism between G and its dual’s dual G^^:

Theorem - The map G⟢G^^ defined by s↦s^^, where s^^⁒(Ο•):=ϕ⁒(s), is a natural isomorphism between G and G^^.

6 Applications

The study of dual groups allows one to visualize Fourier series, Fourier transformsMathworldPlanetmath and discrete Fourier transforms from a more abstract and unified view-point, providing the for a general definition of Fourier transform. Thus, dual groups and Pontryagin duality are the of the of abstract abelian harmonic analysis.

Title Pontryagin duality
Canonical name PontryaginDuality
Date of creation 2013-03-22 17:42:42
Last modified on 2013-03-22 17:42:42
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 7
Author asteroid (17536)
Entry type Theorem
Classification msc 43A40
Classification msc 22B05
Classification msc 22D35
Synonym Pontrjagin duality
Synonym Pontriagin duality
Related topic DualityInMathematics
Defines Pontryagin dual
Defines Pontrjagin dual
Defines Pontriagin dual
Defines dual of an abelian group
Defines character